A certified basis of 4 identities
This list of identities is a basis, proved in Lean, of 265 semigroups of order six, all of which generate the variety V[3301]. Removing the identities that follow from the others leaves 2 identities (Vampire); the variety page shows the shortest known basis.
Identifier sb-c578344cdc4bcd91 in the data of the bases-min seal.
Certified for
[6, 3301] [6, 3317] [6, 3441] [6, 3455] [6, 3504] [6, 3520] [6, 3653] [6, 3663] [6, 3703] [6, 3721] [6, 3766] [6, 3780] [6, 6031] [6, 6043] [6, 6117] [6, 6131] [6, 6134] [6, 6148] [6, 6192] [6, 6240] [6, 6326] [6, 6335] [6, 6338] [6, 6349] [6, 6399] [6, 6411] [6, 6413] [6, 6428] [6, 6480] [6, 6507] [6, 6509] [6, 6544] [6, 6664] [6, 6716] [6, 7310] [6, 7318] [6, 7335] [6, 7506] [6, 7514] [6, 7519] [6, 7520] [6, 7536] [6, 7621] [6, 7622] [6, 7630] [6, 7669] [6, 7670] [6, 7743] [6, 7749] [6, 7759] [6, 7857] [6, 7863] [6, 7867] [6, 7868] [6, 7880] [6, 7892] [6, 7900] [6, 7910] [6, 8062] [6, 8070] [6, 8075] [6, 8076] [6, 8092] [6, 8123] [6, 8135] [6, 8159] [6, 8186] [6, 8217] [6, 8327] [6, 8328] [6, 8334] [6, 8361] [6, 8362] [6, 8378] [6, 8399] [6, 8424] [6, 8425] [6, 8433] [6, 8476] [6, 8477] [6, 8617] [6, 8625] [6, 8648] [6, 8654] [6, 8665] [6, 8673] [6, 8715] [6, 8729] [6, 8739] [6, 9815] [6, 9886] [6, 9923] [6, 10039] [6, 10102] [6, 10106] [6, 10492] [6, 10498] [6, 10500] [6, 10537] [6, 10543] [6, 10549] [6, 10564] [6, 10582] [6, 10588] [6, 10590] [6, 10591] [6, 10597] [6, 10627] [6, 10628] [6, 10661] [6, 10692] [6, 10696] [6, 10702] [6, 10706] [6, 10717] [6, 10736] [6, 10992] [6, 10999] [6, 11002] [6, 11045] [6, 11060] [6, 11061] [6, 11067] [6, 11087] [6, 11090] [6, 11112] [6, 11133] [6, 11139] [6, 11141] [6, 11145] [6, 11238] [6, 11239] [6, 11246] [6, 11277] [6, 11283] [6, 11284] [6, 11289] [6, 11398] [6, 11403] [6, 11404] [6, 11406] [6, 11418] [6, 11422] [6, 11427] [6, 11428] [6, 11437] [6, 11531] [6, 11534] [6, 11536] [6, 11541] [6, 11544] [6, 11650] [6, 11656] [6, 11677] [6, 11684] [6, 11699] [6, 11728] [6, 12534] [6, 12538] [6, 12543] [6, 12585] [6, 12589] [6, 12592] [6, 12593] [6, 12601] [6, 12607] [6, 12647] [6, 12653] [6, 12672] [6, 12676] [6, 12680] [6, 12684] [6, 12694] [6, 12698] [6, 12715] [6, 12719] [6, 12727] [6, 12741] [6, 12745] [6, 13065] [6, 13069] [6, 13072] [6, 13073] [6, 13085] [6, 13088] [6, 13091] [6, 13098] [6, 13101] [6, 13105] [6, 13106] [6, 13110] [6, 13113] [6, 13116] [6, 13121] [6, 13143] [6, 13157] [6, 13211] [6, 13212] [6, 13216] [6, 13231] [6, 13232] [6, 13239] [6, 13257] [6, 13258] [6, 13439] [6, 13440] [6, 13444] [6, 13459] [6, 13460] [6, 13470] [6, 13480] [6, 13481] [6, 13483] [6, 13484] [6, 13492] [6, 13540] [6, 13541] [6, 13544] [6, 13623] [6, 13624] [6, 13628] [6, 13629] [6, 13630] [6, 13636] [6, 13643] [6, 13659] [6, 13660] [6, 13737] [6, 13738] [6, 13739] [6, 13740] [6, 13743] [6, 13894] [6, 13898] [6, 13903] [6, 13945] [6, 13949] [6, 13952] [6, 13953] [6, 13961] [6, 13978] [6, 13979] [6, 13983] [6, 13998] [6, 13999] [6, 14093] [6, 14102] [6, 14127] [6, 14128] [6, 14129] [6, 14143] [6, 14144] [6, 14145] [6, 14147] [6, 14156] [6, 14160] [6, 14162] [6, 14244] [6, 14264] [6, 14343] [6, 14345] [6, 14346] [6, 14348] [6, 14419] [6, 14421]
The identities
- x² ≈ x³
- xy ≈ xy²
- x²y ≈ xyx
- xyz ≈ xzy